The use of the heat operator in an existence theory problem of the Vlasov equation. (English) Zbl 0595.76126

Wollman, S.

Transp. Theory Stat. Phys. 14, 567-593 (1985).

(Author’s summary.) A use is demonstrated for a parabolic operator in the context of the existence theory problem of the Vlasov-Poisson systems. It is shown that some aspects of the evolution of the field can be viewed as a diffusion process and that relevant estimates on the field can be derived through the use of the heat operator. The application of this to the existence theory problem is demonstrated.

Reviewer: J.Blum

MSC:

76X05	Ionized gas flow in electromagnetic fields; plasmic flow
82D10	Statistical mechanics of plasmas
35Q99	Partial differential equations of mathematical physics and other areas of application
46N99	Miscellaneous applications of functional analysis

Keywords:

existence theory; Vlasov-Poisson systems; diffusion process; heat operator

References:

[1]	Bardos C., Global existence for the Vlasov–Poisson equation in 3 space variables with small initial data · Zbl 0593.35076
[2]	Batt J., J. Differential Equations 25 pp 342– (1977) · Zbl 0366.35020
[3]	Glassey R., Contemporary Mathematics 28 pp 269– (1984) · Zbl 0556.35122
[4]	Horst E., Math. in the Appl. Sci. 3 pp 229– (1981)
[5]	Horst E., Math. Meth. in the Appl. Sci. 4 pp 19– (1982) · Zbl 0485.35079
[6]	Reed M., Methods of Mathematical Physics 11 (1975) · Zbl 0308.47002
[7]	Ukai S., Osaka, J. Math. 15 pp 245– (1978)
[8]	Van Kampen N. A., The or etica Methods in Plasma Physics (1967)
[9]	Wollman S., Pure Appl. Math. 33 (1980) · Zbl 0437.45023
[10]	Wollman S., J. Differential Equation 35 (1) pp 30– (1980) · Zbl 0402.76089
[11]	Wollman S., J. Math. Anal. Appl. 90 pp 138– (1982) · Zbl 0506.45012
[12]	Wollman S., ”Existence and Uniqueness Theory of the Vlasov Equation” (1982) · Zbl 0506.45012